[exim] Problems with outgoing messages from virtual domains

I apologize if I'm repeating a well-known problem. I've searched for
possible
solutions but haven't been able to find any. I guess I'm not using the
right
search terms.

I have an Exim + Dovecot virtual domain configuration. My users are defined
in
Dovecot. Exim requires authentication before allowing any users to send
emails to external addresses. The users log in with their full email
addresses
for the username (first.last@??? for example).

My exim.conf contains the following:

primary_hostname = mail.domain-x.com
domainlist local_domains = virtual1.com : virtual2.com : domain-x.com

The problem I have is that when a user sends an email from one of the
virtual
domains (first.last@???), Exim sends the following during the SMTP
session when delivering the email:

MAIL-FROM: first.last@???@mail.domain-x.com

Some domains (including Gmail) reject the message because the MAIL-FROM
command does not contain a valid email address.

Is there a way to stop Exim from adding the primary_hostname to the
MAIL-FROM
address?

Thanks.